1. The problem involves converting radians to degrees and understanding central angles in circle sectors. 2. The formula to convert radians to degrees is: $$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$$ This formula is important because it allows us to translate between the two common units for measuring angles. 3. Let's convert the given radian measures to degrees step-by-step: - For $\frac{2\pi}{3}$ radians: $$\frac{2\pi}{3} \times \frac{180}{\pi} = \cancel{\frac{2\pi}{3}} \times \cancel{\frac{180}{\pi}} = \frac{2 \times 180}{3} = 120^\circ$$ - For $\frac{3\pi}{2}$ radians: $$\frac{3\pi}{2} \times \frac{180}{\pi} = \cancel{\frac{3\pi}{2}} \times \cancel{\frac{180}{\pi}} = \frac{3 \times 180}{2} = 270^\circ$$ - For $\frac{7\pi}{6}$ radians: $$\frac{7\pi}{6} \times \frac{180}{\pi} = \cancel{\frac{7\pi}{6}} \times \cancel{\frac{180}{\pi}} = \frac{7 \times 180}{6} = 210^\circ$$ - For $\frac{5\pi}{2}$ radians: $$\frac{5\pi}{2} \times \frac{180}{\pi} = \cancel{\frac{5\pi}{2}} \times \cancel{\frac{180}{\pi}} = \frac{5 \times 180}{2} = 450^\circ$$ - For $\frac{\pi}{2}$ radians: $$\frac{\pi}{2} \times \frac{180}{\pi} = \cancel{\frac{\pi}{2}} \times \cancel{\frac{180}{\pi}} = \frac{180}{2} = 90^\circ$$ - For $\frac{3\pi}{4}$ radians: $$\frac{3\pi}{4} \times \frac{180}{\pi} = \cancel{\frac{3\pi}{4}} \times \cancel{\frac{180}{\pi}} = \frac{3 \times 180}{4} = 135^\circ$$ 4. These conversions help us understand the shaded sectors in the circles: - 90° corresponds to $\frac{\pi}{2}$ radians (3A circle) - 60° corresponds to $\frac{\pi}{3}$ radians (U2B circle) - 120° corresponds to $\frac{2\pi}{3}$ radians (6c circle and the single shaded circle by problem 2) - 180° corresponds to $\pi$ radians (2d circle) - 270° corresponds to $\frac{3\pi}{2}$ radians (7 circle) - 360° corresponds to $2\pi$ radians (5f circle) This understanding allows us to interpret the central angles and their corresponding radian and degree measures in the circle sector graphs. Final answer: The conversion formula is $$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$$ and the given radian measures convert to degrees as follows: - $\frac{2\pi}{3} = 120^\circ$ - $\frac{3\pi}{2} = 270^\circ$ - $\frac{7\pi}{6} = 210^\circ$ - $\frac{5\pi}{2} = 450^\circ$ - $\frac{\pi}{2} = 90^\circ$ - $\frac{3\pi}{4} = 135^\circ$